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The study of the semi-stability of Einstein-Finsler vector bundles

Research Project

Project/Area Number 10640083
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKagoshima University

Principal Investigator

AIKOU Tadashi  KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (00192831)

Co-Investigator(Kenkyū-buntansha) ATSUMI Tsuyoshi  KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20041238)
OHMOTO Toru  KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (20264400)
MIYAJIMA Kimio  KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (40107850)
SHINMORI Shuichi  KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (40226353)
SAKAI Koukichi  KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20041759)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1998: ¥2,200,000 (Direct Cost: ¥2,200,000)
KeywordsEinstein-Finsler bundle / Kahler morphism / Complex Bott connection / Conformal flatness / Projective flatness / Vanishing theorem / Einstein-Finsler構造 / 凖安定性 / 準安定性
Research Abstract

For a holomorphic vector bundle E over a compact Kahler manifold M, we denote by P(E) the projective bundle associated with E. Then P(E) may be considered as a Kahler morphism, and its relative tangent bundle admits a partial connection D called complex Bott connection. This partial connection is uniquely determined from its pseudo Kahler metric. Any pseudo Kahler metric on P(E) determines a convex Finsler structure F on E. Such a Finsler structure is unique up to the multiplication by a positive function on M. We say (E, F) a Einstein-Finsler if the mean curvature of D satisfies the Einstein condition. In this research, we have studied the (semi-) stability in the sense of Mumford (or Mumford-Takemoto) of Einstein-Finsler vector bundles. Our research is divided mainly in three parts:
(1) The vanishing theorem of Bochner type and its applications,
(2) The semi-stability of Einstein-Finsler vector bundle satisfying some conditions,
(3) Projective (or conformal) invariants and projectively flat Finsler structures.
The (semi-) stability of Einstein-Finsler vector bundle is affirmative if it satisfy some conditions. From among our results, we state the following two theorems which are concerned with (semi-) stability.
(1) Let (E, F) be an Einstein-Finsler vector bundle. If (E, F) is modeled on a complex Minkowski space, then (E, F) is semi-stable in the sense of Mumford-Takemoto.
(2) Let E be a holomorphic vector bundle over a compact Rieman surface. Then E is stable in the sense of Mumford if and only if it admits a projectively flat Finsler structure, or equivalently, its projective bundle P(E)→M is a flat Kahler morphism.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (52 results)

All Other

All Publications (52 results)

  • [Publications] T. Aikou: "A partial connection on complex Finsler bundles and its applications"Illinois J. Math.. 42. 481-492 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Aikou: "Conformal flatness of complex Finsler structures"Publ. Math. Debrecen. 54. 165-179 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Aikou: "Some remarks on the conformal equivalence of complex Finsler structures"in Finslerian Geometries: A Metting of Minds, Kluwer. 35-52 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Aikou: "Some remarks on Finsler vector bundles"Publ. Math. Debrecen. (in press). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Miyajima: "CR construction of the flat deformations of normal isolated singularities"J. Alg. Geom.. 8. 403-470 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Miyajima: "A note on the closed rengeness of vector bundle-valued tangential Cauchy-Riemann complex"in Analysis and Geometry in several Complex Variables. 193-208 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Akabori and K. Miyajima: "A smoothness of deformations of normal strongly pseudo-convex CR-structures"Kyushu J. Math.. 52. 418-438 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ohmoto: "On characteristic classes of fibres/images of mappings"京大数理研講究録 「特異点論とその応用」. (in press). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Atsumi: "Short proofs of Hiramine's results on character values"Osaka J. Math.. (in press).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Atsumi: "Another proof of Hiramine's theorem on three-dimensional Schur rings"数理解析研究所講究録. 1109. 101-105 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Sakai: "On grapfs having exact three vertices with the same degree"Rep. Fac. Sci. Kagoshima Univ.. 31. 1-7 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Sakai: "Ongenerative nembers"Rep. Fac. Sci. Kagoshima Univ.. 32. 1-8 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Sakai and K. Horata: "Some remarks on degree sets of graphs"Rep. Fac. Sci. Kagoshima Univ.. 32. 9-14 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Isokawa and K. Sakai: "Shape of the apollonian packing in the Eucidean plane"Rep. Fac. Sci. Kagoshima Univ.. 31. 9-17 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ohi and S. Shinmori: "A definition of generalized k-out-of-n multistate systems and their structural and probabilistic properties"Japan J. of Industrial and Applied Math.. 15. 263-277 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Koide, S. Shinmori and H. Ishuii: "The evaluations on lower bounds of all-terminal reliability by arc packings for general networks"The Transactions of the IEICE. E82-A,5. 784-789 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Aikou: "A partial connection on complex Finsler bundles and its applications"Illinois J. Math.. 42. 481-492 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Aikou: "Conformal flatness of complex Finsler structures"Pulb. Math. Debrecen. 54. 165-179 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Aikou: "Some remarks on the conformal equivalence of complex Finsler structures"Finslerian Geometries: A Metting of Minds, Kluwer Acad. Publ.. 35-52 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles"Publ. Math. Debrecen. 57(in press). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Miyajima: "CR construction of the flat deformations of normal isolated singularities."J. Alg. Geom.. 8. 403-470 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Miyajima: "A note on the dosed rengeness of vector bundle-valued tangential Cauchy-Riemann complex."Analysis and Geometry in Several Complex Variables. 193-208 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Akahori and K.Miyajima: "A smoothness of deformations of normal strongly pseudo-convex CR-structures."Kyushu J. Math.. 52. 418-438 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ohmoto: "On characteristics of fibres/images of mappings"RIMS Kokyu-roku. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Atsumi: "Short proofs of Hiramine's results on character values"Osaka J. Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Atsumi: "Another proof of Hiramine's theorem on three-dimensional Schur rings"RIMS Kokyu-roku. 1109. 101-105

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Sakai: "On graphs having exact three vertices with the same degree."Rep. Fac. Sci. Kagoshima Univ.. 31. 1-7 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Sakai: "On generative numbers"Rep. Fac. Sci. Kagoshima Univ.. 32. 1-8 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Sakai and K.Horata: "Some remarks on degree sets for graphs"Rep. Fac. Sci. Kagoshima Univ.. 32. 9-14 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Isokawa and K.Sakai: "Shape of the apollonian packing in the Euclidean plane"Rep. Fac. Sci. Kagoshima Univ.. 31. 9-17 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] F.Ohi and S.Shinmori: "A definition of generalized k-out-of-n multistate systems and their structural and probabilistic properties"Japan J. of Industrial and Applied math.. 15. 263-277 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Koide, S.Shinmori and H.Ishii: "The evaluations on lower bounds of all-terminal reliability by arc packings for general networks"The Transactions of the IEICE. E82-A,5. 784-789 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Aikou: "Conformal flatness of complex Finsler structures"Publ.Math.Debrecen. 54. 165-179 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Aikou: "Some remarks on the conformal equivalence of complex Finsler structures"in Finslerian Gemetries:A Metting of Minds,Kluwer.. 35-52 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles"Publ.Math.Debrecen. (in press). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 宮嶋公夫: "CR construction of the flat deformations of normal isolated singularities"J.Alg.Geom.. 8. 403-470 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Miyajima: "A note on the closed rengeness of vector bundle-valued tangential Cauchy-Riemann complex"in Analysis and Geometry in Several Complex Variables. 193-208 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Ohmoto: "On Characteristic Classes of Fibers/Images of Mappings"京大数理研講究録「特異点論とその応用」. (in press). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Atsumi: "Another proof of Hiramine's theorem on three-dimensional Schur rings"京都大学数理解析研究講究録. 1109. 101-105 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Koide,S.Shinmori and H.Ishii: "The Evaluations on Lower Bounds of All-terminal Reliability by Arc Packings for General Networks"The Transactions of the IEICE. 82-A. 784-789 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Sakai: "On generative numbers"Rept.Fac.Sci.Kagoshima Univ.. 32). 1-8 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Sakai and K.Horata: "Some remarks on degree sets for graphs"Rept.Fac.Sci.Kagoshima Univ.. 32. 9-14 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Aikou: "A partial connection on complex Finsler bundles and its applications" Illinois J.Math.42. 481-492 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Aikou: "Conformal Flatness of complex Finsler structures" Publ.Math.Debrecen. (in press). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Aikou: "Some remarks on Finsler vector bundles" Publ.Math.Debrecen. (in press). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Akahori and K.Miyajima: "A smoothness of deformations of normal strongly pseudo-convex CR-structures" Kyushu J.of Math.52. 413-438 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 宮嶋公夫: "強擬凸CR構造の変形と正規孤立特異点の変形" 数理解析研究所講究録. 1037. 49-62 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Miyajima: "CR construction of the flat deformations of normal isolated singularities" J.of Alg.Geom.(in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Atsumi: "A note on equalities in group algebras and character values" 数理解析研究所講究録. 1063. 57-60 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Atsumi: "Short proofs of Hiramine's results on character values" Osaka J.Math.(in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Sakai: "On graphs having exact three vertices with the same degree" Rep.Fac.Sci.Kagoshima Univ.31. 1-7 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Isokawa and K.Sakai: "Shape of the Apollonian packing in the Euclidean plane" Rep.Fac.Sci.Kagoshima Univ.31. 9-17 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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